Representativeness

Representativeness
is the collective term used to describe the following range of
fallacies people make when judging probabilities.

The
problem of base-rate neglect

Insensitivity
to prior probability of outcomes

Insensitivity
to sample size

Misperception
of chance and randomness

We
consider these in turn:

The
problem of base-rate neglect

Consider the
following problem

A
particular heart disease has a prevalence of 1/1000 people. A test
to detect this disease has a false positive rate of 5%. Assume that
the test diagnoses correctly every person who has the disease. What
is the chance that a randomly selected person found to have a
positive result actually has the disease?

This question
was put to 60 students and staff at Harvard Medical School.

Almost half
gave the response 95%.

The average
answer was 56%.

Click here
for the correct answer (and full explanation) that was given by just
11 participants.

Insensitivity
to prior probability of outcomes

Suppose you are
given the following description of a person:

'He
is an extremely athletic looking young man who drives a fast car and
has an attractive blond girlfriend.'

Now answer the
following question:

Is
the person most likely to be a premiership professional footballer
or a nurse?

If you
answered professional
footballer then
you were sucked into this particular fallacy. You made the mistake
of ignoring the base-rate frequencies of the different professions
simply because the description of the person better matched the
stereotypical image. In fact there are only 400 premiership
professional footballers in the UK compared with many thousands of
male nurses, so in the absence of any other information it is far
more likely that the person is a nurse.

The hypothesis
that people evaluate probabilities by representativeness in this way
(thereby ignoring the prior probabilities) was tested by [Kahneman
and Tversky 1973]. Subjects were shown brief personality descriptions
of several individuals, allegedly sampled at random from a group of
100 professionals - all engineers or lawyers. The subjects were asked
to assess, for each description, the probability that it belonged to
an engineer rather than a lawyer. There were two experimental conditions:

1. Subjects
were told the group consisted of 70 engineers and 30 lawyers

2. Subjects
were told the group consisted of 30 engineers and 70 lawyers

The probability
that a particular description belongs to an engineer rather than a
lawyer should be higher in 1 than in 2. However, in violation of Bayes
rule, the subjects produced essentially the same probability
judgements. Subjects were apparently evaluating the likelihood of a
description being an engineer rather than a lawyer by the degree to
which it was representative of the two stereotypes; they were paying
little or no regard to the probabilities of the categories.

When subjects
were given no personality sketch, but were simply asked for the
probability that an unknown individual was an engineer the subjects
correctly gave the responses 0.7 and 0.3 in 1 and 2 respectively.
However, when presented with a totally uninformative description the
subjects gave the probability to be 0.5 in both experiments 1 and 2.

Kahneman &
Tversky concluded that when no specific evidence is given, prior
probabilities are used properly; when worthless evidence is given,
prior probabilities are ignored.

Insensitivity
to sample size

Consider the
problem of two hospitals of different sizes in the same town. In the
large hospital, 45 babies are born each day, whereas only 15 are born
in the smaller hospital. 50% of all babies are boys, but on some days
the percentage will be higher and on other days, it will be lower.

Which hospital
would you expect to record more days per year, when over 60% of the
babies born were boys?

This is an
error of "local" randomness known as "the gambler's
fallacy" or a belief in the "law of small numbers".
People believe that when flipping a coin for example, after several
"heads" the next flip will surely be "tails". The
sequence H-T-H-T-T-H is considered more likely than H-H-H-T-T-T, for
example. It seems to be more random, or it is more representative of
the expected sequence generated by such a random process.

In other words,
the fallacy is that the characteristics of a process will not only be
represented globally in an entire sequence, but also locally in each
of it's parts, and it is a fallacy with which even experienced
researchers frequently expose themselves. For example, by expecting
10 samples to provide statistically significant results in an
experiment or trial, in the same way as if there were 10,000 samples.